How does fuel economy quantitatively depend on altitude for a jet?

Question

We already have a few questions and answers about the qualitative effects of altitude on fuel economy:

• How do I determine the best altitude to fly for fuel economy?
• Why does fuel consumption decrease with increasing aircraft altitude?
• What is the relation between an airplane's altitude and the drag it is experiencing?
• Why do jet engines get better fuel efficiency at high altitudes? (contains formula for thermal efficiency)

I am interested in understanding how the changes in pressure and temperature at different altitudes affect the fuel economy of a turbofan powered aircraft quantitatively. In the end, I would like to make a plot of relative fuel economy vs. altitude that takes all of these effects into account, but I don't know how to quantitatively combine these effects.

Some notes:

• By fuel economy I mean fuel required per distance traveled, not time.
• I am not interested in absolute numbers for the fuel per distance, which would require specifying a particular aircraft. I am only interested in how the fuel per distance figure would relatively change with altitude, e.g. normalized to 1 at sea level.
• I assume ISA (International Standard Atmosphere) profiles for pressure and altitude.
• I assume the no wind case. Different winds at different altitudes will of course have an effect on the result, but it is easy to take this into account after the no wind case is understood.

• Let's assume a typical climb profile for a short- to medium-haul jet airliner: 250/280/0.78

  

  You can see that the TAS increases until reaching Mach 0.78, then decreases due to the lower temperatures causing a lower speed of sound and then remains constant above the tropopause. I am particularly interested in how the fuel economy will behave around these altitudes.

Answer 1

I am adding this community wiki answer to show my current state of research and to provide the plot vs. altitude, which I will update when I learn more. Comments are welcome.

Thermal Efficiency

From this answer by Peter Kämpf, we know that the thermal efficiency for jet engines is given by

$$ \eta = \frac{T_\text{max} - T_\text{amb}}{T_\text{max}} $$

where $ T_\text{amb} $ is just the ambient temperature (from ISA) and $ T_\text{max} $ is the temperature resulting from the combustion. If I understand the answer correctly, this should be about 1100 K above ambient temperature, so I am currently using this term to describe the impact of thermal efficiency on fuel economy:

$$ \epsilon_\text{T} \propto \frac{1}{\eta} = \frac{T_\text{max}}{T_\text{max} - T_\text{amb}} = \frac{T_\text{amb} + 1100 \, \mathrm{K}}{1100 \, \mathrm{K}} $$

I am not sure if the increase in temperature of 1100 K is constant with altitude, so please correct me if this is wrong.

Drag

From another answer by Peter Kämpf, we know that induced drag is proportional to dynamic pressure

$$ q = \frac{v^2}{2} \cdot \rho $$

with $ v $ being the TAS and $ \rho $ the density (known from ISA). Since work required to overcome the drag per distance is proportional to the force, the fuel economy should just scale with

$$ \epsilon_\text{drag} \propto \text{TAS}^2 \cdot \rho $$

Propulsive Efficiency

From this answer we know that the propulsive efficiency for a jet engine is given by

$$ \eta_p = \frac{2}{1 + v_e / v_0} $$

where $v_e$ is the exhaust velocity and $v_0$ is the TAS. As far as I could find, there is no straight-forward way to relate $v_e$ to altitude and temperature. For the moment, I added the propulsive efficiency for a high-bypass jet engine from the following plot from Wikipedia:

^{(image source: Wikipedia)}

Summary

For the combined (relative) fuel economy term, I just multiply all previous terms:

$$ \epsilon = \epsilon_T \cdot \epsilon_\text{Drag} \cdot \epsilon_\text{Prop} $$

The following plot now shows the relative fuel required per distance. Each curve has been normalized to 1 at sea level.

The propulsive efficiency dominates as long as the TAS is increased. Afterwards, the lower drag dominates. The overall fuel use is almost half as low at high cruise altitudes compared to sea level.

Answer 2

I can not see a comprehensive answer to the first question but will answer the next three

1. Why does fuel consumption decrease with increasing aircraft altitude?

Because of an improved ratio of true speed to total drag. See question 2

2. What is the relation between an airplane's altitude and the drag it is experiencing?

With reduced air density more speed is required to create the same dynamic pressure. Both lift and drag are direct functions of the dynamic pressure. The limit on altitude is a combination of Angle of attack and airspeed. Absolute ceiling is determined by gross weight, maximum lift coefficient(high AoA) and maximum dynamic pressure.(most often limited by critical mach and air density) Best altitude is found near the optimum angle of attack ceiling. Which results in just under critical mach and a dynamic pressure where lift=gross weight with an AoA of best lift to drag ratio. This will be below the absolute ceiling because best L/D AoA is generally much lower than the maximum lift AoA.

3. Why do jet engines get better fuel efficiency at high altitudes? (contains formula for thermal efficiency)I will answer

Simply they are designed for best performance at high altitude and artificially throttled at low altitude. Because they are designed to compress cold low density air at high altitude they would overheat or have mechanical stress failures if operated at maximum near sea level. The biggest limit in gas-turbine design is the thermal limits of the material used in the high turbine first stage.(strength at a temperature) Operating below maximum is less efficient, this is a common thermodynamics/entropy engineering problem, the larger the difference in energy the more efficiently that energy can be harnessed(converted to another form) or transferred in or out of a system. This is also a limit in steam engine design, higher temperature boilers and higher pressure steam make them more efficient but material properties and safety are the limits with common heat sources.